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Brain-Computer Interfaces (BCIs) allow users to control devices and communicate by using brain activity only. BCIs based on broad-band visual stimulation can outperform BCIs using other stimulation paradigms. Visual stimulation with pseudo-random bit-sequences evokes specific Broad-Band Visually Evoked Potentials (BBVEPs) that can be reliably used in BCI for high-speed communication in speller applications. In this study, we report a novel paradigm for a BBVEP-based BCI that utilizes a generative framework to predict responses to broad-band stimulation sequences. In this study we designed a BBVEP-based BCI using modulated Gold codes to mark cells in a visual speller BCI. We defined a linear generative model that decomposes full responses into overlapping single-flash responses. These single-flash responses are used to predict responses to novel stimulation sequences, which in turn serve as templates for classification. The linear generative model explains on average 50% and up to 66% of the variance of responses to both seen and unseen sequences. In an online experiment, 12 participants tested a 6 × 6 matrix speller BCI. On average, an online accuracy of 86% was reached with trial lengths of 3.21 seconds. This corresponds to an Information Transfer Rate of 48 bits per minute (approximately 9 symbols per minute). This study indicates the potential to model and predict responses to broad-band stimulation. These predicted responses are proven to be well-suited as templates for a BBVEP-based BCI, thereby enabling communication and control by brain activity only.